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1. Familiar Letters about the Art of Reasoning
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1 1 Familiar Letters about the Art of Reasoning 15 May 1890 Houghton Library Stagira, May 15, 1890. My dear Barbara: The University of Cracow once conferred upon a very good fellow a degree for having taught the philosophical faculty to play cards. I cannot tell you in what year this happened,—perhaps it was 1499. The graduate was Thomas Murner, of whose writings Lessing said that they illustrated all the qualities of the German language; and so they do if those qualities are energy, rudeness, indecency, and a wealth of words suited to unbridled satire and unmannered invective. The diploma of the university is given in his book called Chartiludium, one of the numerous illustrations to which is copied to form the title page of the second book of a renowned encyclopaedia, the Margarita Philosophica . 1 Murner’s pack contained 51 cards. There were seven unequal suits; 3 hearts, 4 clubs (or acorns), 8 diamonds (or bells), 8 crowns, 7 scorpions, 8 fish, 6 crabs. The remaining seven cards were jokers, or unattached to suits; for such cards formed a feature of all old packs. The object of Murner’s cards was to teach the art of reasoning, and a very successful pedagogical instrument they no doubt proved. If you will provide yourself, my dear Barbara, with a complete pack of cards with a joker, 53 in all, I will make a little lesson in mathematics go down like castor-oil in milk. Take, if you will be so kind, the 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 of spades, and arrange these ten cards in their proper order. I mean by this that the ace, or 1, is to be at the back of the pack, the 2 next, and so on, the 10 alone showing its face. I call this the “proper order,” because I propose always to begin the count of cards in 1. Published at Heidelberg in 1496, at Freiberg in 1503, in Strassburg by Grüninger in 1504, in Strassburg by Schott in 1504, in Basle in 1508; etc. Writings of C. S. Peirce 1890–1892 2 a pack at the back, so that, in the pack of ten cards you have just been so obliging as to arrange, every card is in its proper place, that is the number it bears on its face is equal to the number of its place from the back of the pack. The face-value of the 2 nd card is 2, that of the 3 rd card, 3, and so on. Now let us add 3 to the face-value of each card in the pack. How shall we do that without a printing-press? Why, by simply taking 3 cards from the back of the pack of ten and carrying them to the face. The face-value of card number 1 is now 3 ⫹ 1, or 4; that of card 2 is 5, and so on up to card 7 which is 10. Card 8 is 1; but 1 and 11 are the same for us. Since we have only ten cards to distinguish, ten different numbers are enough. We, therefore, treat 1, 11, 21, 31, as equal, because we count round and round the 10, thus: We say 13 and 23 are equal, meaning their remainders after division by 10 are equal. This sort of equality of remainders after division is called congruence by mathematicians and they write it with three lines, thus 13 ⬅ 23 (mod. 10). The number 10 is said to be the modulus, that is, the divisor, or the smallest number congruent to zero, or the number of numbers in the cycle. Instead of ten cards you may take the whole suit of thirteen, and then, imagining a system of numeration in which the base is thirteen and in which we count 1 2 3 4 5 6 7 8 9 10 Jack Queen King 1 2 4 6 7 8 9 3 0 10 21 11 13 14 15 16 17 18 19 12 22 20 -1 5 -2 [44.200.196.114] Project MUSE (2024-03-29 02:04 GMT) 1. Familiar Letters about Reasoning, 1890 3 we have a similar result. Fourteen, or king-ace, is congruent with 1; fifteen , or king-two, with 2, etc. It makes no difference how many cards there are in a pack. To cut it, when arranged in its proper order, and transpose the two parts, is to add...